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We show that there exist non-unitarizable groups without nonabelian free subgroups. Both torsion and torsion free examples are constructed. As a by-product, we show that there exist finitely generated torsion groups with nonvanishing first  L  2 -Betti numbers. We also relate the well-known problem of whether every hyperbolic group is residually finite to an open question about approximation of  L  2 -Betti numbers.

L 2 -Betti Numbers and Non-Unitarizable Groups without Free Subgroups